Your employment contract will be with the local entity of Remote where you are based. For example, if you are based in the United Kingdom your employment contract will be with Remote's UK entity.
Remote employs you locally as the employer of record on behalf of the company you work for and this also means that your employment contract will be with Remote. All the business operations and day-to-day of your work is handled by the company you work for. In order to do this, Remote takes on all of the responsibilities, obligations and tasks related to your employment.